Readings

Welcome to PhiMath 2024! In our first meeting for this academic year, we will discuss a paper by Linnebo and Shapiro on the notion of Potential Infinity.

For our final meeting on structuralism, we cover the modal set-theoretic structuralism from Parsons and Linnebø. This rounds out the book we have been following and also our time with structuralism (for now).

For our next meeting on structuralism, we will be covering the two views championed by the authors of the textboock: Shapiro’s ante rem, or Sui Generis, structuralism and Hellman’s Modal Structuralism.

Main Readings

Chapter 5 and 6 in Hellman, G., & Shapiro, S. (2018). Mathematical Structuralism (Elements in the Philosophy of Mathematics). Cambridge: Cambridge University Press. doi:10.1017/9781108582933

For our second meeting on Structuralism in mathematics, we will be focusing on the category-theoretical side of the discussion. Stemming from work in algebraic topology, category theory has since the 1950’s become an indisposable tool for mainstream mathematics and is seen by some to encode the structuralist philosophy into mathematics itself. We will be reading and discussing some thoughts around the plausibility of using category-theoretic foundations for mathematics and whether this really follows the structuralist philosophy.

We are now covering the theory of structuralism in mathematics. This first meeting will be an overview of the general position and its various parts together with specific focus on set-theoretic structuralism. Over the course of the coming meetings, we will spend some time on various conceptions of structuralism and their proponents.

We are continuing our discussion about Higher-Order Logic (HOL), this time focusing specifically on arguments against adopting Higher-Order logics for various purposes. These arguments range from (HOL) being set theory in disguise, the (lack of) applicability of (HOL) for its intended purposes, and the serious metalogical issues the logic faces.

The reading group in the philosophy of mathematics is back and we hold the first meeting on the 12th of October. The topic will be Higher-Order Logic. We will base the discussion around Stewart Shapiro’s chapter “Higher-order logic” in The Oxford Handbook of Philosophy of Mathematics and Logic.

This time we take a recourse to mereology! In his talk, Elias Bronner will introduce the philosophical motivations for the field, Leśniewski’s nominalistic position. Further, we will discuss basic mereology and its relation to Boolean Algebras. Elias will conclude with touching upon a recent interpretation of mereology within set theory by J.D. Hamkins which suggests that mereology is too weak of a system to serve as a foundation for mathematics.

This time we continued Derek So’s discussion of phenomenology, its recent developments and relations to linear logic and embodied cognition.

Main Readings:

• Tieszen, R. Mathematical Intuition: Phenomenology and Mathematical Knowledge. Dodrecht-Boston-London: Kluwer Academic Publishers.

Additional Readings:

• Zahavi, D. Husserl’s Phenomenology, Stanford Univ. Press, pp. 14-26.

• A fragment on categorial intuition by Jan Gronwald.

In relation to the first chapter of “Mathematical Intuition” by Richard Tiszen, we discussed some notions of Husserl’s phenomenology that pertain to the categorial intuition.

Derek So introduced us to some basic concepts of Husserl’s phenomenology. We focused on intentionality, synthetic and eidetic intuition. Then briefly discussed the different conceptions of intuition in more contemporary phenomenology.

Main Readings:

• Tieszen, R. Mathematical Intuition: Phenomenology and Mathematical Knowledge. Dodrecht-Boston-London: Kluwer Academic Publishers. SUBCHAPTER 2. OF CH. 2, “INTENTIONALITY AND INTUITION”, pp. 21-25.

• Zahavi, D. Husserl’s Phenomenology, Stanford Univ. Press, pp. 14-26.

• A fragment on categorial intuition by Jan Gronwald.